The Dirac Equation for Many-Electron Systems

THE DIRAC EQUATION FOR MANY-ELECTRON SYSTEMS

BY K. S. VISWANATHAN, F.A.SC. (Memoir No. 122 of the Raman Research Institute, Bangalore-6) Received June 11, 1960

1. INTRODUCTION

THE well-known relativistic wave equation for an electron' was derived by Dirac by factorising a second order equation into two linear equations. It is the object of the present note to point out that the Dirac equation is simply the eigenvalue equation of the magnitude of the momentum four- vector, and that one can derive it by expressing the magnitude of the momentum vector in terms of its four components. The above idea enables one to generalise the Dirac equation and obtain a relativistic equation for systems containing several electrons. In Section 3, we have given the wave equation (Equation 16) for a system composed of several particles and this is very similar in form to the Dirac equation. Relativistic wave equations for a system of two electrons have previously been given by Eddington, 2 Gaunt 3 and Breit' of which the one given by Breit is the most satisfactory. By replacing the velocities vl and v" of the electrons by the spin matrices — cal and — call in a Hamiltonian given by Darwin, 4 Breit was able to obtain an approximate wave equation for two-electron systems. It is shown in Section 4, that Equation (16) leads to the Breit equation when it is represented in the product space of the two electrons.

2. THE MOMENTUM FOUR-VECTOR Before proceeding further, we first state a result which was first proved by Weyl6 for a vector in a n-dimensional space and which we shall apply presently.

(a) Lemma.—Let (x1, x2 , • .., x,,) be the co-ordinates of a vector r in an Eucledian space with reference to a system of orthogonal axes, and let r = (x1 2 + x22 + ... + xn 2)' denote the 'magnitude' or 'length' of the vector r. Then 35 36 K. S. VISWANATHAN

r = E (v^xx w=^ ), (1) where the y's are elements of an abstract algebra satisfying the relations (y,y,+yyµ) = 2S, . The y's can be expressed as matrices and for the case n = 4, they are the Dirac matrices.

(b) The Dirac Equation.—Let x 1 , x2, x3 and x4 (= ict) be the co-ordinates of a world point, and similarly let pl, P2, p3 and p4 (= iE/c) denote the components of the momentum four-vector. Besides x 4 and p4i we shall also use the symbols xo and po given by x4 = ixo and p 4 = ip o. Expressed as operators we have then p 4 = — ihZ/^x 4 and po = i^i^/^xo. Now the momentum four-vector P is a vector with constant magnitude imoc where mo is the rest mass of the particle. Apply now the result (1) to the vector P = (pl, P2, P3, p4). We then get

4 E yip = imoc. (2)

If ) is an eigenstate of the momentum four-vector, we get from (2) the equation of an electron as

3 iy4po + z yip — imoc) ) = 0. (3) i=1 For the y's we now choose the following representations :-

I 0 O — ivlO — la2 Y4= Y1=C ig1 0 ), v2—_ i0 0 I); 20),

0 — 1O3 Y3 - i93 O ) where I, a,, v2 and a3 denote respectively the unit matrix in two dimensions and the three Pauli matrices.

Multiplying (3) to the left by — 1V4 1' we get

3 (loo + j' aipi + J3moc 1 0) = 0(4) which is the Dirac equation in its conventional form. When the electron is moving in a field, the components of the momentum four-vector are given by (pi + e/c Ai) (i = 0, 1, 2, 3) where Ao, Ai , A2 , A3 The Dirac Equation for Many-Electron Systems 37

4 are the scalar and vector potentials of the field. From the relations ^ (pi + e/c Ai) 2 = — m02c2, we see that the magnitude of the momentum four- vector is equal to imoc in this case also. Thus by replacing pi by (pi + e/c Ai) in (4), we get the equation for an electron moving in field as

3 ee j(PO + e Ao l + ai (pi + e Ai) + flmoc 10) = 0. (5) The left-hand side of (2) is similar in form to the expression of the length of a vector in terms of its direction cosines. The i's can thus be regarded as the representations in a matrix algebra of the direction cosines of the momentum four-vector. We have thus

Yi= — Pz .^ or iyi= ui, (6 ) IPI where ui (i = 1, 2, 3, 4) are the components of the velocity four-vector. The matrices iyi thus represent the components of the velocity four-vector. Since

ai= — i(Y4)-1 Yi(i=1, 2 ,3), we have

ai= _ZPi = _xi P4 c or

Xi = — Cai. (7) Similarly

2 ^3 = — 1 — 2(8) We thus get the well-known expressions for the components of the velocity of the particle without calculating the commutation relations of x1 , x2 and x 3 with the Hamiltonian. In finding out the dynamical variables that are the classical analogues of products (or quotients) of matrices, care should be taken to verify that only the algebraical rules that are common to both the matrix and the ordi- nary algebras are used. We have derived (4) by multiplying (2) by — iy4 1 making use of the relation (y 4)_' Y 4 = 1 which is common to both algebras, 38 K. S. VISWANATHAN

though one could equally well derive (4) by multiplying (2) by — iy4. The

classical analogue of a2 is thus — i (y 4) - ' yi and not — iy 4 yz.

3. SYSTEM OF MANY ELECTRONS We have seen that the Dirac equation can be derived by expressing the magnitude of the four-vector P + e/c A in terms of its components. Now

(Pi + Ati) = mocui (i = 1, 2, 3, 4)

mo z^, (9) / v/1 2, C2

where u1 , u 2i u 3 , u, are the components of the velocity four-vector. natural way to generalise the Dirac equation for a system of particles would be to consider a four-vector whose components are respectively E mocuz (i = 1, 2, 3, 4, or 0) here the sum is to be taken over all the particles'of the system. We shall denote the components of this vector by (Po + e/c A0), (P1 + e/c A,), (P2 + e/c A2) and (P3 + e/c A3). Now in the special theory of relativity the components of the total momentum are given by 7

P i = — -' f TIkdSk + Z rflocuz, c (10) where Tzk.. are the components of the energy-momentum tensor. In three- dimensional form, we can write for the total momentum of field plus charges

f dV+Ep (10 a) and for the energy f W dV + 2 , (10 b) where

S = C ExH is the Poynting vector and

W = I (E" 2 + H2) The Dirac Equation for Many-Electron Systems 39 is the density of the field energy. For a system of charges, the electrostatic energy (U), apart from the self-energy terms, is equal to

U= f WdV=E eAeB . rnB We thus see that the quantities

— e (A0, Ai, A2, A3) stand respectively for the energy-momentum of the field.

Let us denote the magnitude of P + e(c A by iP'. P' is equal to E moic only as a first approximation. We have in fact s

3 ti — P' 2 = — c$ (^' mi) z -f- ^ ( J mixik) $, (11) o k=1 i-1 where m0 nit -

Thus

- p, = — 2(,E Mj2 + 2 £ miml ,, r 3<1

+(,E mi2via + 2 mimivi . vi +

P12= E mi2c$ (1 — Vie, + 2 E mimlc2 ( i — V c ) ' (12) Now

mimics 1 —v C2 )

=moi oic2 i_! ) J7ii 2 \( C21 _ vf\ J ( 2 . moimojc 1 + 2 2c2)' (13) 40 K. S. VISWANATHAN where vij = (vi — vi). (14) Substituting (13) in (12), we get

P' 2 _ Em0i2c2 + 2 Em0im0jc 2 (1 +v 2 \

_ (E m0ic)2 + E m0im0jvij 2. Thus a second approximation for P' is given by n Emoimojvij P' ^ 2' moic + i- "(15) ' moic Applying now the results of Section 1 to the vector P + e/c A, we get the equation for the system of electrons as

3 (Po + A0 ) + ak (Pk + Ak) + 9P' I ) =0. (16) J k=1

4. REPRESENTATION IN THE PRODUCT SPACE AND THE BREIT EQUATION Equation (16) describes the system as a whole and can be considered to be the equation of motion for the centre of inertia of the system. Since we are operating in a four-dimensional space, the matrices ak and P are all four-dimensional. They refer to the entire-system of electrons and do not contain any labels of the individual particles. In practice, however, one needs equations that bring in explicitly the positions and spins of the individual electrons of the system. We shall see presently that Equation (16) can be transformed into one that contains explicit reference to the spins of the electrons if it is represented in the product space of the electrons. Since the y's represent the direction cosines of the momentum four- vector, their classical analogues are given by n ^' mizik

= ¢ = 1(k = 1, 2, 3) Yk iP and (E imic) Y4 = p, (17 a) The Dirac Equation for Many-Electron Systems 41 Thus we have E mixik ak = — i (Y^)-1 Yk = — c E mi and (17 b) pp , _—P'2cEmi Thus we have

3 , L ak (Pk + c Ak)+flp

k=1

_ l kL' (2' m2 tik) 2 — P12 1 c E mi i =— Af mic= —c f moi — C2 V(I )

^ w 2 _ — moic 1 — c2 — C E miv42(18) icl V i=1

correct to terms of the order of v 4/c3 .

Alternatively, one can get (18) without using the idea of direction cosines at all. From (16) we have

ak (Pk + Ak) + PP S 1 0) = — (Po + L A0) k=i and by definition,

— Cl'o+^A,) — mic; i this leads now to the right-hand of (18).

For an electron moving in a field, we have

my = P + 42 K. S. VISWANATHAN where P is the momentum conjugate to the position of the particle and A is the vector potential of the field. Thus we have

U mzv'2 = U v? . (Pi + e At) (19) The term e/c E vi • Ai gives the interaction energy of the charges with the magnetic field produced by the motion of the electrons. Now the potentials arising from the motion of a charge e have been worked out in Landau and Lifshitz' and these are given by

q' =e, A,_e w+(v' n)n), (20a) r 2cr where r is the distance of the charge from the field point and n = r/r. When there are several charges we must sum over all the charges. Thus the potentials acting at the position of charge 1 due to the motions of charges 2, 3, • • •, n are given by

rlj t=2 and

A' = e ^ wj + (V'I l (20 b) 2crIj e=2 Let us suppose that our system consists of n electrons moving in a static electric field V (r) (the field of the nuclei). We can write the potential energy of the system as

U= — e =—e27c d=1 where

= V (e)r - 21Z' rij ' I The factor I in the above expression for cZ is introduced to take care that the interaction energy between two electrons is not counted twice. The energy is thus expressible as the sum of n different terms, each term standing for the energy density of a particle. Consider now ¢1. We have The Dirac Equation for Many-Electron Systems 43

=V(r,)-2 rl;^ f this can be interpreted as the potential at the field point of electron 1 due to the static field V (r) and the fields of (n — 1) moving charges, each having a charge of — e/2. Thus the vector potential Al that arises as a consequence of the motion of electrons 2, 3, ..., n can be obtained by multiplying (20 b) by — 1. We have

Ai = — e r [vj + (vj ' nil) n1](21) 4cr1j

In general we have

Ai = — e[vL + (vl - nij) nij] 4cril

Substituting (21) in (19) we get

E mivi2 vi • Pi t=1 = Ad=1 _ e 2 r [v nip) (v n3)] rid (22) if i

8 ak (Pk + e Ak) + PP'

kri

2 _ — moic \/l _ v$ — C vi Pi

[v v. + e$ + (vi - rig) (s i •(23) 2c rij rid `f We have seen that the velocity of the electron is related to the spin matrices by means of the relations V=— cap 44 K. S. VISWANATHAN and

^=— 1—c2. V v2 Let us therefore introduce a set of matrices ai and 9i by means of the relations

Vi= — Cat and

162=,J1—v.2C2 (24) A representation for the matrices ai and P in the product space of the electrons can be obtained by adopting the following rule 8 : If two physical systems a and b are compounded to form a total system c, then the system space H of c is R x G where R is the system space of a and G of b. In the system c obtained by composition, a Hermitean form Ax I 2 is associated with a quantity a of a and 1 1 x B with P of b where A and B are the forms associated with a, P in R, G respectively, and I l and I2 are the unit forms in R and G. In the product space, therefore, the matrices ai and P are given by ai = IXIXI... XaXI... xI and Pi = IXIxI... xJxI... XI. (25) In the above, the product contains n terms, and a and P occur in the i-th place; further the x denotes direct product multiplication. Substituting (25), (24) and (23) in (16), we get the equation for the system as

n S

\P0 + C A0) + i^ l.J ak2Pk1 + m0C "1=1 1 i=1 k=1

7 . r2i) e 2at - a? (a2. rij) (a & =O. + 2C rij +7 rij3 (26 ) ii ii i

1. Dirac, P. A. M. .. Principles of Quantum Mechanics, Third Edition, 1947, Chapter XI. 2. Eddington, A. S. .. Proc. Roy. Soc., 1929, 122 A, 358. 3. Gaunt, J. A. .. Ibid., 1929, 122 A, 513. 4. Darwin, C. G. .. Phil. Mag., 1920, 39, 537. 5. Breit, G. .. Phy. Rev., 1929, 34, 553; 1932, 39, 616. 6. Weyl, H. .. The Classical Groups, Princeton, New Jersey, Princeton University Press, 1946. pp. 270-73. 7. Landau, L. and .. The Classical Theory of Fields, Addison—Wesley Press, Lifshitz, E. 1951, pp. 59, 86, 180-85. 8. Weyl, H. .. The Theory of Groups and Quantum Mechanics, Methuen & Co., Ltd., 1931, pp. 89-92.